Burner Knobs are extremely hot to the touch

Strip it down behind the knob and the venturi tube. You have spiders nests back there and the flame is pooling there instead of firing down the venturi tube to the burner. The venturi mixes air with the gas.

There is an O-ring on the shaft that turns to control the flame height it can get old & get to hard or crack letting gas leak out this gas can start burning behind the knob making it very hot.
Some O-rings can be replaced on some grills. If not you'll need to replace the whole knob unit. Or it may be time for a new grill. The knob is made of bakelite that's why it dosen't melt or burn.

Can this grill be also used on propane? If so you might be running it too hot on natural gas, check to see if the control knobs have two different set of markings.

you are missing the heate plate that stops the knobs from getting hot go to ebay and put your model in and look for heate plate or some thing like that

oven seal may need to be replaced

get in a gas fitter as there is a burner problem
probably best if you junk it and get a new unit
much safer with gas units

More than likely, the vinturi valve on each burner is clogged and the burners flames are back firing and reaching the control knobs. Stop using the grill, until you disassemble the burners, clean them and the venturi valves, The venturi is an opening on the sides of the burner, where it slides of each burner orifice. Spiders like to build nest in them.

For a more visual look at what I'm talking about, go to the follwing link for Sure Fire Heat. They manufactured the grill for Amana. Go down the list of Amana grills until you find yours or one like it. In the Owners Manual there should be a section on Maintenance, which will show you how to remove the burners and clean them.

http://www.sureheat.com/manuals.html

Hope this helps you solve the problem.

gas leak on control knobs or from burner

You should test the wiring to see if there is any short that is causing the overheating.
If one knob only is getting hot, then the knob is probably faulty and need to be replaced.

I the all knobs are overheating because of heat coming from the grill, then is a bad heat insulation problem, better to call the service number listed on the owner manual and discuss the problem with Jenn Air in this case.

Here you find replacement parts for your model, in case you need to replace a knob.

Our success has been to heat the elements to hot for about 5 minutes. Then let cool to reset temp sensor.

Can cause the problem by turning a burner on for a short period of time